Properties

Label 12-1470e6-1.1-c1e6-0-0
Degree $12$
Conductor $1.009\times 10^{19}$
Sign $1$
Analytic cond. $2.61557\times 10^{6}$
Root an. cond. $3.42607$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s − 3·9-s + 6·11-s + 6·16-s − 6·19-s − 24·29-s + 9·36-s − 18·41-s − 18·44-s + 24·59-s + 12·61-s − 10·64-s + 36·71-s + 18·76-s − 48·79-s + 6·81-s + 12·89-s − 18·99-s − 48·101-s − 36·109-s + 72·116-s + 15·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 3/2·4-s − 9-s + 1.80·11-s + 3/2·16-s − 1.37·19-s − 4.45·29-s + 3/2·36-s − 2.81·41-s − 2.71·44-s + 3.12·59-s + 1.53·61-s − 5/4·64-s + 4.27·71-s + 2.06·76-s − 5.40·79-s + 2/3·81-s + 1.27·89-s − 1.80·99-s − 4.77·101-s − 3.44·109-s + 6.68·116-s + 1.36·121-s + 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(2.61557\times 10^{6}\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.01795303646\)
\(L(\frac12)\) \(\approx\) \(0.01795303646\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{3} \)
3 \( ( 1 + T^{2} )^{3} \)
5 \( 1 - 4 p T^{3} + p^{3} T^{6} \)
7 \( 1 \)
good11 \( ( 1 - 3 T + 6 T^{2} - 17 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 - 45 T^{2} + 1107 T^{4} - 17390 T^{6} + 1107 p^{2} T^{8} - 45 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 30 T^{2} + 927 T^{4} + 900 p T^{6} + 927 p^{2} T^{8} + 30 p^{4} T^{10} + p^{6} T^{12} \)
19 \( ( 1 + 3 T + 45 T^{2} + 90 T^{3} + 45 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 81 T^{2} + 2979 T^{4} - 75606 T^{6} + 2979 p^{2} T^{8} - 81 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 + 12 T + 120 T^{2} + 720 T^{3} + 120 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( ( 1 + 78 T^{2} - 10 T^{3} + 78 p T^{4} + p^{3} T^{6} )^{2} \)
37 \( 1 - 165 T^{2} + 12387 T^{4} - 566110 T^{6} + 12387 p^{2} T^{8} - 165 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 + 9 T + 75 T^{2} + 610 T^{3} + 75 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{3} \)
47 \( 1 - 9 T^{2} + 1659 T^{4} - 160614 T^{6} + 1659 p^{2} T^{8} - 9 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 - 51 T^{2} + 8814 T^{4} - 284551 T^{6} + 8814 p^{2} T^{8} - 51 p^{4} T^{10} + p^{6} T^{12} \)
59 \( ( 1 - 12 T + 150 T^{2} - 980 T^{3} + 150 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 - 6 T + 75 T^{2} - 20 T^{3} + 75 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 150 T^{2} + 16167 T^{4} - 1267540 T^{6} + 16167 p^{2} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 - 6 T + p T^{2} )^{6} \)
73 \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{3} \)
79 \( ( 1 + 24 T + 414 T^{2} + 4194 T^{3} + 414 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 300 T^{2} + 40392 T^{4} - 3707850 T^{6} + 40392 p^{2} T^{8} - 300 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 - 6 T + 159 T^{2} - 676 T^{3} + 159 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 240 T^{2} + 16752 T^{4} - 546370 T^{6} + 16752 p^{2} T^{8} - 240 p^{4} T^{10} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.21221671930387086648674689568, −4.85854121421563609386384187848, −4.65010320904731160907007833926, −4.52666682955703933651577200749, −4.07697810550151471510627519791, −4.06461009041190486666230887583, −3.98330725366455158673658874402, −3.97744239849485532273953921276, −3.81510560408775911598319439865, −3.76803277727488799508832031825, −3.44136828798755727750582068112, −3.30753054084456600091580964765, −3.11594778067769678181324130957, −2.85719719758762009061822757496, −2.64097155902423886063811277505, −2.36621599225756956351554951738, −2.18813479494393730126419119963, −2.18610881993371807543755290364, −1.83133442830073518017343496669, −1.51650479241496825950742900242, −1.37956142411026077863953869654, −1.21748140205341073717931451469, −0.931331034842372190810524805246, −0.30251969546203392118981620802, −0.03747268631843508602171277101, 0.03747268631843508602171277101, 0.30251969546203392118981620802, 0.931331034842372190810524805246, 1.21748140205341073717931451469, 1.37956142411026077863953869654, 1.51650479241496825950742900242, 1.83133442830073518017343496669, 2.18610881993371807543755290364, 2.18813479494393730126419119963, 2.36621599225756956351554951738, 2.64097155902423886063811277505, 2.85719719758762009061822757496, 3.11594778067769678181324130957, 3.30753054084456600091580964765, 3.44136828798755727750582068112, 3.76803277727488799508832031825, 3.81510560408775911598319439865, 3.97744239849485532273953921276, 3.98330725366455158673658874402, 4.06461009041190486666230887583, 4.07697810550151471510627519791, 4.52666682955703933651577200749, 4.65010320904731160907007833926, 4.85854121421563609386384187848, 5.21221671930387086648674689568

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.